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A unified formula of the optimal portfolio for piecewise hyperbolic absolute risk aversion utilities

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  • Zongxia Liang
  • Yang Liu
  • Ming Ma
  • Rahul Pothi Vinoth

Abstract

We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utilities, including many classic and non-standard utilities as examples. A typical application is the composition of a HARA preference and a piecewise linear payoff in asset allocation. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. We conduct a general asymptotic analysis to the optimal portfolio, which directly serves as an analytical tool for financial analysis. We compare this PHARA portfolio with those of other utility families both analytically and numerically. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability of the PHARA utility. Finally, we use financial data to test the performance of the PHARA portfolio in the market.

Suggested Citation

  • Zongxia Liang & Yang Liu & Ming Ma & Rahul Pothi Vinoth, 2024. "A unified formula of the optimal portfolio for piecewise hyperbolic absolute risk aversion utilities," Quantitative Finance, Taylor & Francis Journals, vol. 24(2), pages 281-303, January.
    • Handle: RePEc:taf:quantf:v:24:y:2024:i:2:p:281-303
    DOI: 10.1080/14697688.2023.2300664
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    Cited by:

    1. Shanyu Han & Jian Lei & Yang Liu, 2024. "Equilibrium in Style: A Modeling Framework on the Cash Flow and the Life Cycle of a Consumer Store," Papers 2404.02426, arXiv.org, revised May 2024.

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